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Babylonian method
Using the Babylonian method you can approximate the value of a positive square root x = √a by iterations of
Explanation
We want to calculate x = √a and search for an equivalent equation to do so
We can make an iteration of this, where √a always lies between xn and a / xn. Then you can assume that the average of these two values gives a better value, so that
Example 1
If we start for x = √2 with a = 2 and x0 = 2, we get the following iterations
That's even more accurate than the value 1,414213562… your calculator shows.
Example 2
If we start for x = √9 with a = 9 and x0 = 2, we get the following iterations
If we start with x0 = 3, then the result is immediately
because √9 = 3, but we already knew that.